Question

Write out the first five terms of the sequence an=(−2)n
. Determine if the sequence is arithmetic or geometric, and then find the common ratio or difference. Which option below is accurate?


Option #1: The sequence is arithmetic, and the common difference is 6.

Option #2: The sequence is arithmetic, and the common difference is −2
.

Option #3: The sequence is geometric, and the common ratio is −2
.

Option #4: The sequence is geometric, and the common ratio is 2.

(1 point)
Option #
is accurate.

Answer 1

Let's first calculate the first five terms of the sequence given by \( a_n = (-2)^n \):

1. For \( n = 1 \): \( a_1 = (-2)^1 = -2 \)
2. For \( n = 2 \): \( a_2 = (-2)^2 = 4 \)
3. For \( n = 3 \): \( a_3 = (-2)^3 = -8 \)
4. For \( n = 4 \): \( a_4 = (-2)^4 = 16 \)
5. For \( n = 5 \): \( a_5 = (-2)^5 = -32 \)

So the first five terms of the sequence are:
- \( a_1 = -2 \)
- \( a_2 = 4 \)
- \( a_3 = -8 \)
- \( a_4 = 16 \)
- \( a_5 = -32 \)

Now let's determine if the sequence is arithmetic or geometric.

**For arithmetic sequences**, the difference between consecutive terms (the common difference) is constant. We can calculate the differences:

- \( a_2 - a_1 = 4 - (-2) = 4 + 2 = 6 \)
- \( a_3 - a_2 = -8 - 4 = -12 \)
- \( a_4 - a_3 = 16 - (-8) = 16 + 8 = 24 \)
- \( a_5 - a_4 = -32 - 16 = -48 \)

Since the differences are not constant, the sequence is **not arithmetic**.

**For geometric sequences**, the ratio of consecutive terms (the common ratio) is constant. We can calculate the ratios:

- \( \frac{a_2}{a_1} = \frac{4}{-2} = -2 \)
- \( \frac{a_3}{a_2} = \frac{-8}{4} = -2 \)
- \( \frac{a_4}{a_3} = \frac{16}{-8} = -2 \)
- \( \frac{a_5}{a_4} = \frac{-32}{16} = -2 \)

Since the ratios are constant, the sequence is **geometric** with a common ratio of \( -2 \).

Thus, the correct answer is:

**Option #3: The sequence is geometric, and the common ratio is −2.**